Francesc Prats

Using consensus and distances between generalized multi-attribute linguistic assessments for group decision-making

This paper proposes a mathematical framework and methodology for group decision-making under multi-granular and multi-attribute linguistic assessments. It is based on distances between linguistic assessments and a degree of consensus. Distances in the space of qualitative assessments are defined from the geodesic distance in graph theory and the Minkowski distance. The degree of consensus is defined through the concept of entropy of a qualitatively-described system. Optimal assessments in terms of both proximity to all the expert opinions in the group and the degree of consensus are used to compare opinions and define a methodology to rank multi-attribute alternatives.

Measuring consensus in group decisions by means of qualitative reasoning

This paper presents a mathematical framework to assess the consensus found among different evaluators who use ordinal scales in group decision-making and evaluation processes. This framework is developed on the basis of the absolute order-of-magnitude qualitative model through the use of quantitative entropy. As such, we study the algebraic structure induced in the set of qualitative descriptions given by evaluators. Our results demonstrate that it is a weak partial semi-lattice structure that in some conditions takes the form of a distributive lattice. We then define the entropy of a qualitatively described system. This enables us, on the one hand, to measure the amount of information provided by each evaluator and, on the other hand, to consider a degree of consensus among the evaluation committee. This new approach is capable of managing situations where the assessment given by experts involves different levels of precision. In addition, when there is no consensus regarding the group decision, an automatic process assesses the effort required to achieve said consensus.

Relative and absolute order-of-magnitude models unified

The aim of this paper is to analyze under which conditions Absolute Order-of-Magnitude and Relative Order-of-Magnitude models may be concordant and to determine the constraints which guarantee concordance. A graphical interpretation of the constraints is provided, bridging the absolute qualitative labels of two quantities into their corresponding relative relation(s), and conversely. The relative order of magnitude relations are then characterized in the absolute order-of-magnitude world.

Modeling group assessments by means of hesitant fuzzy linguistic term sets

Hesitant linguistic term sets have been introduced to capture the human way of reasoning using linguistic expressions involving different levels of precision. In this paper, a lattice structure is provided to the set of hesitant fuzzy linguistic term sets by means of the operations intersection and connected union. In addition, in a group decision making framework, hesitant fuzzy linguistic descriptions are defined to manage situations in which decision makers are assessing different alternatives by means of hesitant fuzzy linguistic term sets. Based on the introduced lattice structure, two distances between hesitant fuzzy linguistic descriptions are defined. These metric structures allow distances between decision makers to be computed. A centroid of the decision making group is proposed for each distance to model group representatives in the considered group decision making framework.

A Qualitative Reasoning Approach to Measure Consensus

This chapter introduces a mathematical framework on the basis of the absolute order-of-magnitude qualitative model. This framework allows to develop a methodology to assess the consensus found among different evaluators who use ordinal scales in group decision-making and evaluation processes. The concept of entropy is introduced in this context and the algebraic structure induced in the set of qualitative descriptions given by evaluators is studied. We prove that it is a weak partial semilattice structure that in some conditions takes the form of a distributive lattice. The definition of the entropy of a qualitatively-described system enables us, on one hand, to measure the amount of information provided by each evaluator and, on the other hand, to consider a degree of consensus among the evaluation committee. The methodology presented is able of managing situations where the assessment given by experts involves different levels of precision. In addition, when there is no consensus within the group decision, an automatic process measures the effort necessary to reach said consensus.

Using L-fuzzy sets to introduce information theory into qualitative reasoning

We formally construct the extended set of qualitative labels L over a well-ordered set. The qualitative descriptions of a given set are defined as L-fuzzy sets. In the case where the well-ordered set is finite, a distance between L-fuzzy sets is introduced based on the properties of the lattice L. The concept of the information contained in a qualitative label is introduced, leading to a formal definition of the entropy of an L-fuzzy set as a Lebesgue integral. In the discrete case, this integral becomes a weighted average of the information of the labels, corresponding to the Shannon entropy in information theory.

Analysis of Tensions in a Population Based on the Adequacy Concept

The concept of similarity between objects has traditionally been taken as the criterion for recognising their membership of a given class. This paper considers how well an object fits into a class by using the concept of adequacy introduced by the LAMDA learning system [6],[9]. The Global Adequacy Degree (GAD) is a function of the objects class membership. An adequacy threshold is associated with a non-informative class (NIC). Objects falling below this threshold value are not considered to belong to any significant class. In this research, the tensions produced by a classification scheme are defined by means of the adequacy of an object in a class. This allows us to analyse the stability orbalance of the scheme. An example is given in the form of the adequacy and the tension of a classification scheme for a group of customers patronising an imaginary shop.

Using Orders of Magnitude and Nominal Variables to Construct Fuzzy Partitions

The application of qualitative reasoning to learning algorithms can provide these models with the capability of automate common-sense and expert reasoning. Learning algorithms aim at automatically gathering the relevant information from a set of patterns and turn it into useful knowledge. That information usually comes from different sources and displays subjectivity and ambiguity, especially as far as qualitative data is concerned. This paper analyses the unsupervised learning capability of the LAMDA (learning algorithm for multivariate data analysis) algorithm. The LAMDA algorithm relies on the generalising capability of fuzzy connectives obtained as the interpolation of a t-norm and its dual t-conorm and permits the use of qualitative variables. Qualitative variables defined on orders of magnitude scales or on nominal scales are used to reduce the search space. A mathematical property of the hybrid connectives used is imposed to guarantee coherence in the obtained classification. The results obtained are applied to support decision making in a marketing problem: identifying customer behaviour.

Kernel Functions over Orders of Magnitude Spaces by Means of Usual Kernels. Application to Measure Financial Credit Risk

This paper lies within the domain of learning algorithms based on kernel functions, as in the case of Support Vector Machines. These algorithms provide good results in classification problems where the input data are not linearly separable. A kernel is constructed over the discrete structure of absolute orders of magnitude spaces. This kernel will be applied to measure firms’ financial credit quality. A simple example that allows the kernel to be interpreted in terms of proximity of the patterns is presented.

L-fuzzy sets for group linguistic preference modeling: An application to assess a firm's performance

Given a finite totally ordered set of linguistic descriptions, the extended set of qualitative labels with different levels of precision L is constructed. In this framework, qualitative descriptions of a given set are L-fuzzy sets. A distance between L-fuzzy sets is introduced based on the properties of the lattice L. An illustrative example in the retail sector applied to assess a firms overall performance using perceptions of managers in the firms different departments is presented.